Harish G. Ramaswamy

Harikrishna Narasimhan, Harish G. Ramaswamy, Shiv Kumar Tavker et al.

We present consistent algorithms for multiclass learning with complex performance metrics and constraints, where the objective and constraints are defined by arbitrary functions of the confusion matrix. This setting includes many common performance metrics such as the multiclass G-mean and micro F1-measure, and constraints such as those on the classifier's precision and recall and more recent measures of fairness discrepancy. We give a general framework for designing consistent algorithms for such complex design goals by viewing the learning problem as an optimization problem over the set of feasible confusion matrices. We provide multiple instantiations of our framework under different assumptions on the performance metrics and constraints, and in each case show rates of convergence to the optimal (feasible) classifier (and thus asymptotic consistency). Experiments on a variety of multiclass classification tasks and fairness-constrained problems show that our algorithms compare favorably to the state-of-the-art baselines.

Lakshmi Narayan Pandey, Rahul Vashisht, Harish G. Ramaswamy

Attention mechanisms form a core component of several successful deep learning architectures, and are based on one key idea: ''The output depends only on a small (but unknown) segment of the input.'' In several practical applications like image captioning and language translation, this is mostly true. In trained models with an attention mechanism, the outputs of an intermediate module that encodes the segment of input responsible for the output is often used as a way to peek into the `reasoning` of the network. We make such a notion more precise for a variant of the classification problem that we term selective dependence classification (SDC) when used with attention model architectures. Under such a setting, we demonstrate various error modes where an attention model can be accurate but fail to be interpretable, and show that such models do occur as a result of training. We illustrate various situations that can accentuate and mitigate this behaviour. Finally, we use our objective definition of interpretability for SDC tasks to evaluate a few attention model learning algorithms designed to encourage sparsity and demonstrate that these algorithms help improve interpretability.

Rahul Vashisht, Harish G. Ramaswamy

Attention models are typically learned by optimizing one of three standard loss functions that are variously called -- soft attention, hard attention, and latent variable marginal likelihood (LVML) attention. All three paradigms are motivated by the same goal of finding two models -- a `focus' model that `selects' the right \textit{segment} of the input and a `classification' model that processes the selected segment into the target label. However, they differ significantly in the way the selected segments are aggregated, resulting in distinct dynamics and final results. We observe a unique signature of models learned using these paradigms and explain this as a consequence of the evolution of the classification model under gradient descent when the focus model is fixed. We also analyze these paradigms in a simple setting and derive closed-form expressions for the parameter trajectory under gradient flow. With the soft attention loss, the focus model improves quickly at initialization and splutters later on. On the other hand, hard attention loss behaves in the opposite fashion. Based on our observations, we propose a simple hybrid approach that combines the advantages of the different loss functions and demonstrates it on a collection of semi-synthetic and real-world datasets

P. Krishna Kumar a, Harish G. Ramaswamy

Theoretical studies on the representation power of GNNs have been centered around understanding the equivalence of GNNs, using WL-Tests for detecting graph isomorphism. In this paper, we argue that such equivalence ignores the accompanying optimization issues and does not provide a holistic view of the GNN learning process. We illustrate these gaps between representation and optimization with examples and experiments. We also explore the existence of an implicit inductive bias (e.g. fully connected networks prefer to learn low frequency functions in their input space) in GNNs, in the context of graph classification tasks. We further prove theoretically that the message-passing layers in the graph, have a tendency to search for either discriminative subgraphs, or a collection of discriminative nodes dispersed across the graph, depending on the different global pooling layers used. We empirically verify this bias through experiments over real-world and synthetic datasets. Finally, we show how our work can help in incorporating domain knowledge via attention based architectures, and can evince their capability to discriminate coherent subgraphs.

Harish G. Ramaswamy, L. A. Prashanth

We consider the problem of estimating and optimizing utility-based shortfall risk (UBSR) of a loss, say $(Y - \hat Y)^2$, in the context of a regression problem. Empirical risk minimization with a UBSR objective is challenging since UBSR is a non-linear function of the underlying distribution. We first derive a concentration bound for UBSR estimation using independent and identically distributed (i.i.d.) samples. We then frame the UBSR optimization problem as minimization of a pseudo-linear function in the space of achievable distributions $\mathcal D$ of the loss $(Y- \hat Y)^2$. We construct a gradient oracle for the UBSR objective and a linear minimization oracle (LMO) for the set $\mathcal D$. Using these oracles, we devise a bisection-type algorithm, and establish convergence to the UBSR-optimal solution.

Rahul Vashisht, P. Krishna Kumar, Harsha Vardhan Govind et al.

Neural networks trained with stochastic gradient descent exhibit an inductive bias towards simpler decision boundaries, typically converging to a narrow family of functions, and often fail to capture more complex features. This phenomenon raises concerns about the capacity of deep models to adequately learn and represent real-world datasets. Traditional approaches such as explicit regularization, data augmentation, architectural modifications, etc., have largely proven ineffective in encouraging the models to learn diverse features. In this work, we investigate the impact of pre-training models with noisy labels on the dynamics of SGD across various architectures and datasets. We show that pretraining promotes learning complex functions and diverse features in the presence of noise. Our experiments demonstrate that pre-training with noisy labels encourages gradient descent to find alternate minima that do not solely depend upon simple features, rather learns more complex and broader set of features, without hurting performance.

Umangi Jain, Harish G. Ramaswamy

Despite their massive success, training successful deep neural networks still largely relies on experimentally choosing an architecture, hyper-parameters, initialization, and training mechanism. In this work, we focus on determining the success of standard gradient descent method for training deep neural networks on a specified dataset, architecture, and initialization (DAI) combination. Through extensive systematic experiments, we show that the evolution of singular values of the matrix obtained from the hidden layers of a DNN can aid in determining the success of gradient descent technique to train a DAI, even in the absence of validation labels in the supervised learning paradigm. This phenomenon can facilitate early give-up, stopping the training of neural networks which are predicted to not generalize well, early in the training process. Our experimentation across multiple datasets, architectures, and initializations reveals that the proposed scores can more accurately predict the success of a DAI than simply relying on the validation accuracy at earlier epochs to make a judgment.

Depen Morwani, Rahul Vashisht, Harish G. Ramaswamy

Recent papers have shown that sufficiently overparameterized neural networks can perfectly fit even random labels. Thus, it is crucial to understand the underlying reason behind the generalization performance of a network on real-world data. In this work, we propose several measures to predict the generalization error of a network given the training data and its parameters. Using one of these measures, based on noise resilience of the network, we secured 5th position in the predicting generalization in deep learning (PGDL) competition at NeurIPS 2020.

Depen Morwani, Harish G. Ramaswamy

We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. We analyse both standard weight normalization (SWN) and exponential weight normalization (EWN), and show that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate. We extend these results to gradient descent, and establish asymptotic relations between weights and gradients for both SWN and EWN. We also show that EWN causes weights to be updated in a way that prefers asymptotic relative sparsity. For EWN, we provide a finite-time convergence rate of the loss with gradient flow and a tight asymptotic convergence rate with gradient descent. We demonstrate our results for SWN and EWN on synthetic data sets. Experimental results on simple datasets support our claim on sparse EWN solutions, even with SGD. This demonstrates its potential applications in learning neural networks amenable to pruning.

Mingyuan Zhang, Harish G. Ramaswamy, Shivani Agarwal

The F-measure is a widely used performance measure for multi-label classification, where multiple labels can be active in an instance simultaneously (e.g. in image tagging, multiple tags can be active in any image). In particular, the F-measure explicitly balances recall (fraction of active labels predicted to be active) and precision (fraction of labels predicted to be active that are actually so), both of which are important in evaluating the overall performance of a multi-label classifier. As with most discrete prediction problems, however, directly optimizing the F-measure is computationally hard. In this paper, we explore the question of designing convex surrogate losses that are calibrated for the F-measure -- specifically, that have the property that minimizing the surrogate loss yields (in the limit of sufficient data) a Bayes optimal multi-label classifier for the F-measure. We show that the F-measure for an $s$-label problem, when viewed as a $2^s \times 2^s$ loss matrix, has rank at most $s^2+1$, and apply a result of Ramaswamy et al. (2014) to design a family of convex calibrated surrogates for the F-measure. The resulting surrogate risk minimization algorithms can be viewed as decomposing the multi-label F-measure learning problem into $s^2+1$ binary class probability estimation problems. We also provide a quantitative regret transfer bound for our surrogates, which allows any regret guarantees for the binary problems to be transferred to regret guarantees for the overall F-measure problem, and discuss a connection with the algorithm of Dembczynski et al. (2013). Our experiments confirm our theoretical findings.

Anirban Laha, Saneem A. Chemmengath, Priyanka Agrawal et al.

Converting an n-dimensional vector to a probability distribution over n objects is a commonly used component in many machine learning tasks like multiclass classification, multilabel classification, attention mechanisms etc. For this, several probability mapping functions have been proposed and employed in literature such as softmax, sum-normalization, spherical softmax, and sparsemax, but there is very little understanding in terms how they relate with each other. Further, none of the above formulations offer an explicit control over the degree of sparsity. To address this, we develop a unified framework that encompasses all these formulations as special cases. This framework ensures simple closed-form solutions and existence of sub-gradients suitable for learning via backpropagation. Within this framework, we propose two novel sparse formulations, sparsegen-lin and sparsehourglass, that seek to provide a control over the degree of desired sparsity. We further develop novel convex loss functions that help induce the behavior of aforementioned formulations in the multilabel classification setting, showing improved performance. We also demonstrate empirically that the proposed formulations, when used to compute attention weights, achieve better or comparable performance on standard seq2seq tasks like neural machine translation and abstractive summarization.

Harish G. Ramaswamy, Clayton Scott, Ambuj Tewari

Mixture proportion estimation (MPE) is the problem of estimating the weight of a component distribution in a mixture, given samples from the mixture and component. This problem constitutes a key part in many "weakly supervised learning" problems like learning with positive and unlabelled samples, learning with label noise, anomaly detection and crowdsourcing. While there have been several methods proposed to solve this problem, to the best of our knowledge no efficient algorithm with a proven convergence rate towards the true proportion exists for this problem. We fill this gap by constructing a provably correct algorithm for MPE, and derive convergence rates under certain assumptions on the distribution. Our method is based on embedding distributions onto an RKHS, and implementing it only requires solving a simple convex quadratic programming problem a few times. We run our algorithm on several standard classification datasets, and demonstrate that it performs comparably to or better than other algorithms on most datasets.

Harish G. Ramaswamy, Ambuj Tewari, Shivani Agarwal

We consider the problem of $n$-class classification ($n\geq 2$), where the classifier can choose to abstain from making predictions at a given cost, say, a factor $α$ of the cost of misclassification. Designing consistent algorithms for such $n$-class classification problems with a `reject option' is the main goal of this paper, thereby extending and generalizing previously known results for $n=2$. We show that the Crammer-Singer surrogate and the one vs all hinge loss, albeit with a different predictor than the standard argmax, yield consistent algorithms for this problem when $α=\frac{1}{2}$. More interestingly, we design a new convex surrogate that is also consistent for this problem when $α=\frac{1}{2}$ and operates on a much lower dimensional space ($\log(n)$ as opposed to $n$). We also generalize all three surrogates to be consistent for any $α\in[0, \frac{1}{2}]$.

Harish G. Ramaswamy, Harikrishna Narasimhan, Shivani Agarwal

We study consistency of learning algorithms for a multi-class performance metric that is a non-decomposable function of the confusion matrix of a classifier and cannot be expressed as a sum of losses on individual data points; examples of such performance metrics include the macro F-measure popular in information retrieval and the G-mean metric used in class-imbalanced problems. While there has been much work in recent years in understanding the consistency properties of learning algorithms for `binary' non-decomposable metrics, little is known either about the form of the optimal classifier for a general multi-class non-decomposable metric, or about how these learning algorithms generalize to the multi-class case. In this paper, we provide a unified framework for analysing a multi-class non-decomposable performance metric, where the problem of finding the optimal classifier for the performance metric is viewed as an optimization problem over the space of all confusion matrices achievable under the given distribution. Using this framework, we show that (under a continuous distribution) the optimal classifier for a multi-class performance metric can be obtained as the solution of a cost-sensitive classification problem, thus generalizing several previous results on specific binary non-decomposable metrics. We then design a consistent learning algorithm for concave multi-class performance metrics that proceeds via a sequence of cost-sensitive classification problems, and can be seen as applying the conditional gradient (CG) optimization method over the space of feasible confusion matrices. To our knowledge, this is the first efficient learning algorithm (whose running time is polynomial in the number of classes) that is consistent for a large family of multi-class non-decomposable metrics. Our consistency proof uses a novel technique based on the convergence analysis of the CG method.

Harish G. Ramaswamy, Shivani Agarwal

We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures the smallest `size' of a prediction space in which it is possible to design a convex surrogate that is calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, we apply our framework to study various subset ranking losses, and use the convex calibration dimension as a tool to show both the existence and non-existence of various types of convex calibrated surrogates for these losses. Our results strengthen recent results of Duchi et al. (2010) and Calauzenes et al. (2012) on the non-existence of certain types of convex calibrated surrogates in subset ranking. We anticipate the convex calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.